Answer
$x=8$
Work Step by Step
First, the solutions must satisfy $\left\{\begin{array}{l}
x-6\gt 0\\
x+3\gt 0
\end{array}\right.\quad \Rightarrow x\gt 6\qquad (*)$
in order for the equation to be defined.
On the LHS, apply$ \quad\log_{a}(MN)=\log_{a}M+\log_{a}N$
$\ln[ (x-6)(x+3)]=\ln 22$
... apply the principle of logarithmic equality
$(x-6)(x+3)=22$
$x^{2}-3x-18=22$
$ x^{2}-3x-40=0\quad$to factor, find factors of $-40$ with sum $-3$
... these are $-8$ and $+5$
$(x+5)(x-8)=0$
Possible solutions:
$ x=-5\qquad$... does not satisfy (*), not a solution.
$x=8\qquad $... satisfies (*), and is a valid solution.
